I noticed another new AGS game in my nearby Harrah’s Casino, called Premium Hold’Em, extending their products of 3-hole card poker games against the dealer. I kind of enjoy squeezing these hole cards, catching minimal glimpses of colors and symbols, and making (correct) decisions without knowing my full hand. Sometimes, this leaves me with emergency outs after seeing the dealer’s hand.

Plus, once you’ve played enough Ultimate Texas Hold’Em (UTH), all those decisions become automatic, and all you seem to notice is that string of miracle dealer hands which constantly scoop the table. UTH can be boring *and* painful.

So, what’s new with AGS’s Premium Hold’Em? Well, first, there’s the novelty and the squeezing of your three hole cards. Next, the all-in preflop raise is lowered to 3x, which should reduce the variance a bit (from UTH’s 4x). Then, the entire community board is dealt out at once as a four card “flop”. Finally, you can either 2x, 1x, or fold after seeing your entire hand! (Compare to UTH’s 2x decision before the turn+river.) Of course, the dealer has three unseen hole cards, but the 2x option is still more favorable than in UTH. Overall, these advantages are offset by the lowered 3x preflop raise, and the higher pair-of-treys Dealer qualifier. Bottom lime, the house edge of Premium Hold’Em is lower than UTH, at 2.06% of an Ante.

Oh, and the bonus side bet pays down to Jacks Up (two pairs), and has a low house edge of 4.87%. Again, this offers lower variance than the UTH Trips side bet.

Basic Strategy

I wanted to try out the game, so I worked out a Basic Strategy. I worked it down to the detail that I like, using hand features that make sense to me. The table is organized by the two decision points, and divided into mutually exclusive categories for the Player hand, and also for the board. For each determined sub-category, the selected betting rules should be followed top-down, looking for the first matched condition, or else falling through to the bottom action.

The betting rules are meant to be concise and unambiguous, but I’ll elaborate on some of them below, to clarify with examples.

The Basic Strategy is sub-optimal by only 0.28% from the ideal 2.06% house edge.

Typical play is actually pretty easy, and the Basic Strategy is intuitive and easy to learn. The top part of the table tells you what the 3x preflop raising hands are. They occur about 15.7% of the time, and are easy to remember.

All the river decisions for hands Three-of-a-Kind and higher are simple and intuitive. Thankfully, you should 2x bet *all* full houses and flushes, independent of the board. The only time you slow down with a straight is when you 1x it against a 4-flush board. Notice you still 2x the idiot end straight against an open-ended board, and you still 2x a baby flush against a 4-flush board. And you still 2x a flush against a double-paired board.

You’ll slow down with Three-of-a-Kind when they’re on board. You will 2x them with a nut (best) kicker in the hole. For example, say you’re holding Kh 9s 6h, and there’s trips on board with 8s 8c 8h Ad. You have the nut (best) kicker in the hole (Kh), so you can still 2x your hand. However, if your hole cards were Qh 9s 6h, you should only 1x the hand since your Qh is only 2nd nut kicker. You could still 1x call with Jh 9s 6h (3rd nut kicker), but you would fold with anything lower than the 3rd nut kicker.

If you have Three-of-a-Kind and the board is a scare flush (4-flush), then you can still 1x your hand. For example, if your hole cards were 7d 3s 3h and the board was 3c Kc Jc 6c, you’d only 1x your trip 3’s.

You’ll generally 2x your two pairs, unless the board is dangerous. So, against a scare flush (4-flush) or a scare straight (open-ended) board, you’ll just 1x your two pairs. You should 2x raise your two pairs any time you can beat a double-paired board, or even if you just have the nut kicker. If the board is paired, you’ll 2x raise your top pair or over pair. You can 2x raise with bottom pair, just as long as you have 3rd nut kicker or better. You’ll never fold two pairs when you beat the board. The only time you’ll fold with two pairs is when you’re playing the board, and don’t have 4th nut kicker or better.

Note: “bottom pair” on a paired board means you’ve paired the lowest singleton on the board. For example, say the board is 3s 3h Kd 7h. If you’re holding a 7, you’ve made bottom pair on the board. If you’re holding a K, you’ve made top pair on the board. If you have a pair of wired Aces, they’d be an over pair to the board. If you have a pair of wired Deuces, they’d be an under pair to the board. Also, if you held a pair of wired Fives in the hole, they’d be an under pair to the board too (lowest board singleton is a Seven).

There are a few occasions when you can 2x raise one pair against a safe board (no 4-flush or 4-straight board, including gutshot board). Against a safe board, you can 2x raise one pair if it’s an over pair, top pair, or second pair.

Generally speaking, you’ll at least 1x call any pair made with the board. You will fold 3rd or bottom pair against a scare board. You’ll only call a wired under pair to the board if it’s qualifying, and there’s no 4-flush or 4-straight on board.

You can 1x call playing one pair on board, if you have the nut kicker. You can also 1x call the pair on board if you hold the 2nd nut kicker, and the board is 2-flush or less.

There’s only one case to 1x bet a no-pair hand on the river. You should 1x play an Ace-King high hand against a rainbow flop when holding the nut kicker, and the board isn’t 4-cards to a straight. For example, if your hole cards were Qh 8d 3s and the board was Ac Kh 9d 4s, then you should 1x your hand, because your Qh hole card is the nut kicker.

Collusion Analysis

The game is not vulnerable to even 5 confederates colluding at a full-table sharing perfect information. Ideal decisions using all known player hole cards (15 of them at a full table) yield only about a +1.4% improvement over Basic Strategy (about +0.5% from counter-strategy pre-flop decisions, and about +0.9% from counter-strategy river decisions).

Bonus Side Bet

An optional bonus on your final hand, or the Dealer’s final hand, is available before the hand begins. The table below shows the payouts and their frequencies and returns. The house edge is a reasonable 4.87%, and it pays down to Jacks Up (two pairs), instead of the UTH Trips or better bonus.

Outcome

Combinations

Frequency

Payout

Return

ROYAL_FLUSH

4,324

0.000032

50

0.001616

STRAIGHT_FLUSH

37,260

0.000279

30

0.008355

FOUR_OF_A_KIND

224,848

0.001681

10

0.016807

FULL_HOUSE

3,473,184

0.025961

6

0.155766

FLUSH

4,047,644

0.030255

4

0.121020

STRAIGHT

6,180,020

0.046194

3

0.138581

THREE_OF_A_KIND

6,461,620

0.048299

2

0.096597

Two Pairs (J’s Up+)

173,854,08

0.129951

1

0.129951

other

95,970,252

0.717349

-1

-0.717349

total

133,784,560

1.000000

-0.048656

expected

133,784,560

Detailed Stats

The total outcomes for every possible starting hand, for every possible flop, and every possible dealer hole cards are shown in the table below, following optimal decisions. The player will 3x about 15.7% of his hands, will 2x about 37.4% of his hands, will 1x about 23.9% of the hands, and will fold about 23.0% of the hands.

Ok, I’ll just come out and say it. AGS’s (relatively) new Chase-the-Flush game “Makes Gambling Great Again” 🙂 I actually did the game development math on this a few years ago, when I wasn’t gambling much. But this weekend I discovered a table at my local Hollywood Jamul Casino, so I worked out the Basic Strategy. And guess what? It’s a fun and elegant game! It has a similar structure to Ultimate Texas Hold’Em, with an equal Ante and “Blind” (called the X-Tra Bonus), and 3x/2x/1x Play decisions. But, the game is much more fun, because of the novel flush decisions, and because it’s a lot less frustrating and dread-inducing than the loved/hated UTH. (Hint: the X-Tra pays off more frequently than the Blind, and a 3x pre-flop raise doesn’t miss the board as often as a 4x UTH bet.)

The Rules

The layout below shows the betting spots and payouts of the main game, and the pay table of the optional, independent Same Suit Bonus. The game is played with a standard 52-card deck, where each player and the dealer receives three hole cards and shares a four-card community board to make their best-of-seven flush hand.

The Player wagers an Ante and equal X-Tra Bonus bet before the hand begins. Each Player and the Dealer receive three hole cards. A Player may wager a 3x All-In bet based on his hole cards, or he may check and see the two-card flop. The flop community cards are dealt, and a previously checking Player may now wager a 2x All-In bet, or check again. The final two community board cards are dealt, and a previously checking Player must now wager a 1x All-In bet, or else fold his hand.

Each Player and the Dealer forms the highest flush made from their three hole cards and the four card community board. The Dealer qualifies with a 3-card Nine-high flush or better. If the Dealer doesn’t qualify, the remaining Antes are pushed back to the Player. The qualified Antes and the All-In bet then receive even-money action against the Dealer hand. The Player must beat the Dealer to receive the X-Tra Bonus payout listed in the pay table. If the Dealer’s hand beats the Player’s hand, the X-Tra Bonus loses. All bets push on a tie.

The House Edge

The house advantage for the main Chase-the-Flush game is only 2.65% of an Ante. That’s very reasonable, and is comparable to the UTH house edge. The Basic Strategy yields a practical -2.99% return to the player. The Same Suit Bonus for the pay table in the above layout has a reasonable house edge of 5.67%.

Basic Strategy

I crafted out as simple a Basic Strategy as possible, in terms of how people intuitively view their hands during the game. The following strategy shows the betting conditions for each of the 3x (pre-flop), 2x (flop), and 1x (river) decision points. Check your hand for any of the betting requirements listed per decision point. If your hand doesn’t match any of the listed conditions for the decision point, then you shouldn’t bet it.

For optimal play, you’ll 3x raise about 23.8% of your hands, bet 2x on the flop about 24.9% of the time, 1x call on the river about 35.2% of the time, and otherwise fold about 16.1% of the time.

3x Pre-flop Examples

You should 3x Play any suited Ace. For example, Ac-2c-2d has a EV(3x) of +68.6% and an EV(check pre-flop) of +59.9%. So, it’s still worth +8.7% to 3x raise the hand instead of checking it.

You should 3x any pair of Aces with a Four or better kicker. However, the hand is only +EV for A-A-6 or higher.

You should 3x a suited King with a Six or higher singleton (i.e., the offsuit card). For example, Kd-9d-6c has a EV(3x) of +45.2% and an EV(check pre-flop) of +43.3%, showing it’s slightly better to 3x raise the hand than check it.

You should 3x raise a rainbow hand if each of the ordered ranks are higher than, or equal to, A-Q-T. This means you should 3x A-Q-T, A-K-T, A-A-T, A-Q-J, etc. You should check A-J-T, A-K-6, A-J-J, etc.

2x Flop Examples

The Basic Strategy bets almost all 3-card flushes on the flop. The only exception is when the board is suited, and you’re using a hole card less than a Six to make the flush, AND you don’t have another 2-card flush. Otherwise, you’re betting all other 3-card flushes (or better). For example, say you’re holding Kh-7d-5c and the flop is Ac-2c. You shouldn’t bet your 3-card flush, because your 5c kicker is less than a Six. Notice however, if you were instead holding Kh-7h-5c, you’d 2x Play your 3-card club flush with Five kicker, because you also have a two card Kh-7h flush.

You can 2x bet a 2-card “nut” flush when you have any another 2-card flush. For example, if you’re holding Ac-6d-5h and the flop is 5c 7d, you have the 2-card “nut” (i.e., highest possible) flush in clubs, along with another 2-card flush (7d-6d). You should 2x Play this hand, because one of your hole cards makes the “nut” 2-card flush with a board card, and your hand makes another 2-card flush. Note you shouldn’t bet your Ac-7d-6d hand with a board of 8c-Ah, because many single dealer heart cards (Nine or higher) beat your 2-card flush.

You can also bet two 2-card flushes that use both offsuit board cards with two hole cards averaging a Jack or higher. For example, you can 2x Play your Kh-Qs-2d when the board flops a heart and a spade.

1x River Examples

The Basic Strategy bets almost all 3-card flushes on the river. The only exception is when you’re playing a single small hole card to make your hand, and the board is double-suited. In most of these cases, there are 15 or more single dealer cards that’ll beat your hand.

Otherwise, if the board is rainbow, you’ll always 1x Play any 3-card flush.

If the board has only two cards of one suit, and you have any 3-card flush, there are always less than 15 single dealer cards that’ll beat your hand, so you’ll always 1x play any 3-card flush.

If the board has a 3-card flush on board, you’ll 1x Play the board since Basic Strategy says to always call when there are less than 15 single dealer cards that’ll beat your hand (there are only 10 remaining cards of the flush suit). However, you can get a little fancy, and fold if the board singleton is higher than the 3-card flush AND you don’t hold any cards of the singleton suit.

You can play a very high 2-card flush against a rainbow board if there are less than 10 single dealer cards that’ll beat your hand. This usually means you can play a very high 2-card flush using the highest board card if it’s not paired on board. For example, if the board is 9s-7h-6d-5c, you can 1x Play a Kh in the hole, since the only single dealer cards that will beat your Kh-7h is an As, Ks, Ah, Ad, Ac (5 of them).

Same Suit Bonus

While straight flushes don’t have any meaning in the main game, they are included in the pay table (along with 4+ card regular flushes) in the optional Same Suit Bonus bet. The resulting payouts are very attractive, and add a nice dimension to the game. The breakdown of the 7-card hand outcomes is listed in the table below, and show a total house edge of 5.67% (good as far as bonuses go).

Outcome

Combinations

Frequency

Payout

Return

6-or-7 Card Straight Flush

1,624

0.000012

2000

0.024278

5 Card Straight Flush

39,312

0.000294

100

0.029385

4 Card Straight Flush

636,272

0.004756

20

0.095119

7 Card Flush

6,664

0.000050

300

0.014899

6 Card Flush

256,620

0.001918

50

0.095908

5 Card Flush

3,550,872

0.026542

10

0.265417

4 Card Flush

25,735,424

0.192365

1

0.192365

Nothing

103,557,792

0.774064

-1

-0.774064

Total

133,784,560

1.000000

-0.056694

Optimal Play Statistics

The following table breaks down the total outcomes for the main Chase-the-Flush game, over all possible starting hands, using optimal decisions. The total return in the lower right corner shows a house edge of 2.65% of the Ante.

I saw this new variant of Casino War at Barona Casino, where they player gets an option to swap his card and make a 1x Raise bet. Of course, the catch is the dealer gets two cards, and gets to use the highest one. I wanted to see what the strategy and house edge were, and to check if it was at all countable out of the One-2-Six CSM they use.

The rules are pretty simple. You’re dealt one card face up, and the dealer is dealt two cards face down. The dealer will use his highest card. You have the option to replace your card with the next card out of the shoe (CSM), but you must wager an additional 1x bet to do make this swap. Finally, you may wager an optional 1x Raise on your final hand.

The dealer reveals his hand, and all your bets receive action against the dealer high card. Wins win a Six or lower pay 2:1, else it pays even-money. Ties push, and there’s no “going to War”.

For a 6-deck CSM game, the house edge is a fair 2.56%.

The basic strategy is pretty simple. You should swap an Eight or lower card. You should Raise a Jack or higher final card.

I checked the countability in a CSM by assuming perfect play given 16 known cards before every hand. The EV barely changes by +/- 0.3%, and thus is never +EV.

Ultimate Casino War Optimal Outcomes (6 Decks)

Outcome

Combinations

Frequency

Net

Return

Win 3x bet with drawn A

165,477,312

0.035607

3

0.106820

Win 3x bet with drawn K

138,914,496

0.029891

3

0.089673

Win 3x bet with drawn Q

114,674,112

0.024675

3

0.074026

Win 3x bet with drawn J

92,756,160

0.019959

3

0.059877

Lose 3x bet with drawn card

163,441,152

0.035169

-3

-0.105506

Tie 3x bet with drawn card

97,187,328

0.020912

0

0.000000

Win 2x bet with drawn T

73,160,640

0.015742

2

0.031485

Win 2x bet with drawn 9

55,887,552

0.012026

2

0.024051

Win 2x bet with drawn 8

40,772,160

0.008773

2

0.017546

Win 2x bet with drawn 7

28,274,400

0.006084

2

0.012168

Win 2x bet with drawn 6

18,057,600

0.003886

4

0.015542

Win 2x bet with drawn 5

10,121,760

0.002178

4

0.008712

Win 2x bet with drawn 4

4,466,880

0.000961

4

0.003845

Win 2x bet with drawn 3

1,092,960

0.000235

4

0.000941

Lose 2x bet with drawn card

1,409,976,288

0.303394

-2

-0.606788

Tie 2x bet with drawn card

88,157,160

0.018969

0

0.000000

Win 2x bet with original A

306,488,448

0.065949

2

0.131898

Win 2x bet with original K

257,453,856

0.055398

2

0.110796

Win 2x bet with original Q

212,690,880

0.045766

2

0.091532

Win 2x bet with original J

172,199,520

0.037053

2

0.074107

Lose 2x bet with original card

301,682,880

0.064915

-2

-0.129830

Tie 2x bet with original card

179,437,536

0.038611

0

0.000000

Win 1x bet with original T

135,979,776

0.029260

1

0.029260

Win 1x bet with original 9

104,031,648

0.022385

1

0.022385

Lose 1x bet with original card

409,808,160

0.088181

-1

-0.088181

Tie 1x bet with original card

65,156,976

0.014020

0

0.000000

Total

4,647,347,640

1.000000

-0.0256406

Expected

4,647,347,640

According to Dan Lubin, there’s a version that pays 2:1 for a win with a Six, 3:1 for a win with a Five, 5:1 for a win with a Four, and 8:1 for a win with a Trey. For a 6-deck game, these payouts reduce the house edge to 1.27%. The basic strategy remains the same. Still, the game never gets +EV with only 16 known cards.

DJ Wild is a new “deuces wild” poker game against the dealer, using a standard 52-card deck plus one additional Joker. The game is pretty simple, where you wager an Ante and equal Blind bet before receiving a five card hand. You then decide to either 2x Play the hand, or fold. The Dealer also receives a five card hand, and always qualifies. The Ante and Play bets receive even money action against the Dealer hand, but the Blind only pays for a straight or better. The Blind pays nice odds for rare hands, but only pays about 6% of your hands.

When I first looked at this game, it looked like easy pickings for a table full of colluding advantage players. The confederates would silently share the number of Deuces or Jokers they held in their hands (using simple chip signaling). The whole table would know the number of outstanding Wild cards seen. Each player would 2x Play if they had better than the minimum hand needed for the given Wild count. It looked like the game was toast.

So, I quickly coded it all up to find the theoretical 6-way collusion edge. I was shocked to find that even perfect info sharing only yielded +0.5% between 6 players. You’d expect more of an edge on a perfect 2x Play decision and the always-qualifying Ante. Plus, you get the chance to “save” the Blind bet with a weak hand when the Wild count is high.

Anyways, I worked out a simple 6-way collusion strategy, just in case it turned out to be slighly +EV. The strategy just uses separate minimum calling hands for each Wild count (0 thru 5). The strategy below only decreases the house edge to 1.1%.

6-Way Collusion Strategy for DJ Wild Poker

Wild Count

Minimum Play Hand

0

Pair Jacks

1

Pair Nines

2

Pair Sevens

3

Pair Fours

4

Ace-King high

5

Ace high

Well, at least we know now. No one needs to lose any sleep over this game.

Arizona Stud is a new poker-based table game debuting at the Red Wind Casino in Olympia, WA next week (6 Aug 2014). In this game, both the Dealer and the player each receive three hole cards. The player must discard one of his hole cards before the flop, while the Dealer must use exactly two hole cards to make a hand. After the player discards, he may wager a Play bet of 2x-4x the Ante, or check pre-flop. The two card flop is then revealed, as well as one of the Dealer’s hole cards. If the player checked pre-flop, he must then make a 1x Play bet, or fold. Finally, the community river card and all Dealer hole cards are revealed. The Dealer qualifies with a hand of AK-high or better. The Ante pushes if the Dealer doesn’t qualify. The Play bet always receives even-money action against the Dealer hand.

The set of all possible outcomes for the optimal player is listed in the table below. The total in the lower right corner shows a house edge of 1.34% of the Ante. Note that you should either 4x bet pre-flop, or check. You should never only bet 2x.

Optimal Outcomes for Arizona Stud

Outcome

Combinations

Frequency

Net

Return

Win 4x Play w/ ROYAL_FLUSH against qualified dealer

59,240,916

0.000001

5

0.000005

Win 4x Play w/ FULL_HOUSE against qualified dealer

89,284,476,240

0.001605

5

0.008025

Win 4x Play w/ FLUSH against qualified dealer

10,295,059,284

0.000185

5

0.000925

Win 4x Play w/ STRAIGHT against qualified dealer

13,761,723,420

0.000247

5

0.001237

Win 4x Play w/ THREE_OF_A_KIND against qualified dealer

674,048,087,712

0.012117

5

0.060586

Win 4x Play w/ TWO_PAIRS against qualified dealer

1,233,004,030,272

0.022165

5

0.110827

Win 4x Play w/ ONE_PAIR against qualified dealer

3,533,244,131,304

0.063516

5

0.317580

Win 4x Play w/ HIGH_CARD against qualified dealer

44,095,696,596

0.000793

5

0.003963

Lose 4x Play against qualified dealer

5,323,636,585,296

0.095701

-5

-0.478507

Push 4x Play against qualified dealer

90,869,346,720

0.001634

0

0.000000

Win 4x Play w/ ROYAL_FLUSH against unqualified dealer

20,545,164

0.000000

4

0.000001

Win 4x Play w/ FLUSH against unqualified dealer

4,736,232,972

0.000085

4

0.000341

Win 4x Play w/ STRAIGHT against unqualified dealer

5,959,832,148

0.000107

4

0.000429

Win 4x Play w/ THREE_OF_A_KIND against unqualified dealer

350,836,147,584

0.006307

4

0.025227

Win 4x Play w/ TWO_PAIRS against unqualified dealer

28,557,204,480

0.000513

4

0.002053

Win 4x Play w/ ONE_PAIR against unqualified dealer

2,530,675,447,344

0.045493

4

0.181973

Win 4x Play w/ HIGH_CARD against unqualified dealer

1,146,771,919,728

0.020615

4

0.082461

Lose 4x Play against unqualified dealer

25,248,339,684

0.000454

-4

-0.001816

Push 4x Play against unqualified dealer

25,026,495,696

0.000450

0

0.000000

Win 1x Play w/ ROYAL_FLUSH against qualified dealer

94,841,496

0.000002

2

0.000003

Win 1x Play w/ STRAIGHT_FLUSH against qualified dealer

541,732,152

0.000010

2

0.000019

Win 1x Play w/ FOUR_OF_A_KIND against qualified dealer

6,309,658,080

0.000113

2

0.000227

Win 1x Play w/ FULL_HOUSE against qualified dealer

45,729,841,680

0.000822

2

0.001644

Win 1x Play w/ FLUSH against qualified dealer

51,704,956,552

0.000929

2

0.001859

Win 1x Play w/ STRAIGHT against qualified dealer

90,255,233,808

0.001622

2

0.003245

Win 1x Play w/ THREE_OF_A_KIND against qualified dealer

591,850,723,248

0.010640

2

0.021279

Win 1x Play w/ TWO_PAIRS against qualified dealer

1,068,681,540,840

0.019211

2

0.038423

Win 1x Play w/ ONE_PAIR against qualified dealer

4,443,972,518,832

0.079888

2

0.159776

Win 1x Play w/ HIGH_CARD against qualified dealer

188,023,085,280

0.003380

2

0.006760

Lose 1x Play against qualified dealer

10,848,202,319,420

0.195015

-2

-0.390029

Push 1x Play against qualified dealer

161,798,077,992

0.002909

0

0.000000

Win 1x Play w/ ROYAL_FLUSH against unqualified dealer

14,941,584

0.000000

1

0.000000

Win 1x Play w/ STRAIGHT_FLUSH against unqualified dealer

220,863,744

0.000004

1

0.000004

Win 1x Play w/ FLUSH against unqualified dealer

23,880,526,848

0.000429

1

0.000429

Win 1x Play w/ STRAIGHT against unqualified dealer

48,873,031,380

0.000879

1

0.000879

Win 1x Play w/ THREE_OF_A_KIND against unqualified dealer

34,291,273,536

0.000616

1

0.000616

Win 1x Play w/ TWO_PAIRS against unqualified dealer

301,557,935,124

0.005421

1

0.005421

Win 1x Play w/ ONE_PAIR against unqualified dealer

3,348,961,937,952

0.060203

1

0.060203

Win 1x Play w/ HIGH_CARD against unqualified dealer

2,842,212,690,936

0.051094

1

0.051094

Lose 1x Play against unqualified dealer

1,090,180,312,308

0.019598

-1

-0.019598

Push 1x Play against unqualified dealer

100,426,510,008

0.001805

0

0.000000

folds

15,186,972,477,600

0.273011

-1

-0.273011

total

55,627,620,048,000

1.000000

-0.013402

expected

55,627,620,048,000

The basic strategy for the game is listed in the table below, which returns a 1.70% house edge. The player should 4x his hand about 27% of the time, 1x call about 46% of the time, and fold the remaining 27% of the time.

The game looks like fun. The strategy is actually pretty simple, but you get to make the occasional decision. I’ll actually be in Seattle next week (my first time), so I’ll try to check out the game. Maybe I could hit a nice bad beat for once.

Arizona Stud Basic Strategy

Decision

Strategy

discard

Hold pair, else
hold two highest cards.
Advanced exception: hold highest and lowest cards when suited, AND
highest two cards aren’t suited, AND highest card is Eight or better, AND
middle card is Six or lower, AND lowest card is only one rank below middle
card.

The optional 2 Pair Plus Bonus bet pays for the final hand made by the player. The house edges for the various offered paytables are listed below.

2 Pair Plus Paytables

Player Hand

Paytable A

Paytable B

Paytable C

Paytable D

Royal Flush

500-to-1

500-to-1

500-to-1

500-to-1

Straight Flush

200-to-1

200-to-1

200-to-1

200-to-1

Four-of-a-Kind

100-to-1

100-to-1

100-to-1

80-to-1

Full House

50-to-1

50-to-1

40-to-1

40-to-1

Flush

30-to-1

25-to-1

30-to-1

30-to-1

Straight

20-to-1

20-to-1

20-to-1

20-to-1

Three-of-a-Kind

6-to-1

6-to-1

6-to-1

6-to-1

Two Pairs

4-to-1

4-to-1

4-to-1

4-to-1

others

lose

lose

lose

lose

House Edge

3.03%

5.12%

5.53%

6.58%

The Player Bad Beat Bonus bet pays when a player’s Jacks-or-Better hand is beat by the Dealer. The following table shows the optimal outcomes for the strategy maximizing the Bad Beat Bonus return. The house edge for the optimal Bad Beat Bonus strategy is 8.00%.

I saw a new Hold’Em type game at Casino Pauma last week, and I thought I’d work out the numbers and give it a try. The game is pretty simple. You bet an Ante before the hand begins. After seeing your two hole cards, you may bet 2x preflop, or check. After the flop, you may 1x bet or check. The turn, river, and the dealer’s hole cards are then revealed. The dealer qualifies with a pair of 6’s or better. If the dealer doesn’t qualify, all post-Ante wagers push. If the dealer beats your hand, you lose all your remaining bets. If you beat a qualified dealer hand, you win all your bets. If you beat a non-qualified dealer, you only win 1/2 your Ante.

The game is a bit calmer than Ultimate Texas Hold’Em, since you only have a single Ante, and you can check it down to showdown (in fact, this happens 69.8% of the time). Plus, players may like the fact that they can make the 2x and 1x bets only when they have an advantage. (I.e., all properly made 2x and 1x bets are +EV.) And the Ante is only a -11.4% loser, on average. The optimal player makes a 2x preflop bet 11.2% of the time, and a 1x flop bet on 25.5% of the time. The dealer qualifies 69.1% of the time. The game has relatively low variance, and I found myself increasing the Ante from the $5 minimum, to $10, and $15. (I’d never do that with UTH.)

The total outcomes for the optimal player strategy are listed in the table below, and show a house edge of 3.2% of the Ante.

1 Bet Threat Optimal Outcomes

Outcome

Combinations

Frequency

Net

Return

Bet 2x and 1x and beat qualified dealer

884,580,718,240

0.031804

4

0.127215

Bet 2x and 1x and beat non-qualified dealer

505,981,246,728

0.018192

0.5

0.009096

Bet 2x and 1x and lose to qualified dealer

374,729,986,984

0.013473

-4

-0.053891

Bet 2x and 1x and lose to non-qualified dealer

5,856,935,220

0.000211

-1

-0.000211

Bet 2x and 1x and tie dealer

25,182,150,868

0.000905

0

0.000000

Bet 2x only and beat qualified dealer

293,907,701,760

0.010567

3

0.031701

Bet 2x only and beat non-qualified dealer

387,449,913,432

0.013930

0.5

0.006965

Bet 2x only and lose to qualified dealer

524,307,039,216

0.018851

-3

-0.056552

Bet 2x only and lose to non-qualified dealer

76,858,269,780

0.002763

-1

-0.002763

Bet 2x only and tie dealer

25,553,189,772

0.000919

0

0.000000

Bet 1x only and beat qualified dealer

2,434,367,467,360

0.087524

2

0.175047

Bet 1x only and beat non-qualified dealer

1,467,870,962,280

0.052775

0.5

0.026387

Bet 1x only and lose to qualified dealer

1,215,166,965,412

0.043689

-2

-0.087379

Bet 1x only and lose to non-qualified dealer

17,931,292,692

0.000645

-1

-0.000645

Bet 1x only and tie dealer

164,852,060,176

0.005927

0

0.000000

Bet ante only and beat qualified dealer

3,363,692,256,360

0.120936

1

0.120936

Bet ante only and beat non-qualified dealer

4,003,403,426,760

0.143936

0.5

0.071968

Bet ante only and lose to qualified dealer

9,229,633,097,868

0.331836

-1

-0.331836

Bet ante only and lose to non-qualified dealer

1,896,770,105,748

0.068195

-1

-0.068195

Bet ante only and tie dealer

915,715,237,344

0.032923

0

0.000000

Total

27,813,810,024,000

1.000000

-0.032157

I worked out the basic strategy for the game, just in case anyone wants to play the game. The strategy is actually pretty simple. Since the dealer qualifies with a pair of 6’s or better, you generally only bet the flop if there’s a qualified hand to beat. You can bet kickers and draws against a qualified flop, otherwise you should only bet a qualifying pair when there’s a board card lower than your pair, but 6 or higher.

The basic strategy below has an error rate of 4.5%, that only results in a cost of 0.23% to the player. So the practical house edge is 3.5% for the game.

1 Bet Threat Basic Strategy

Wager

Player Hand

Rules

2x

Pairs

2x bet a pocket pair of 7’s or better, elsecheck pocket 2’s thru 6’s.

Suited

Bet QJs, KTs, KJs, KQs, and A8s or better, elsecheck all others.

Offsuit

Bet KQo, and ATo or better, elsecheck all others.

1x

Straight or better

Always bet.

Three-of-a-Kind

Always bet, except if trips on flop and less than 2nd nut kicker.

Two Pairs

Bet if flop not paired, elsebet if flop qualified (pair 6’s or better), elsebet if board has undercard to pairs, elsebet 9’s up or better, elsecheck all others.

There’s not much opportunity for collusion in the game. Knowledge of the hole cards of all 6 players will modify some of the preflop 2x decisions, but the frequency and value of these counter-(basic)strategy decisions aren’t enough to overcome the 3.2% house edge. Trust me, I’d have worked it out if it was worthwhile.

There’s two bonus bets offered, where the Pocket Bonus pays when your hole cards make a pocket pair, and the Final Hand bonus on your final 7-card hand. The paytables offered at Casino Pauma aren’t very good.

A reader told me about a new ShuffleMaster game at The D Casino, where you try to make a 4- to 7- card flush, starting with 4 hole cards, and paying to see 5th/6th and 7th Streets on a community board. The betting structure is similar to Mississippi Stud, where you post an Ante, then make a 1x Play decision to see 6th Street, and a final 1x Play decision to see 7th Street. The game pays odds on the Ante if you make a 4-card flush or better, and even-money on the 1x Play bets. Otherwise, if you fold or don’t make a hand, you lose your bets.

Ante Pay Table

Length

Flush

Straight Flush

7

300-to-1

1000-to-1

6

20-to-1

500-to-1

5

9-to-1

100-to-1

4

5-to-1

15-to-1

I believe The D will award the highest possible payout for a given hand. So, if you make a 5-card flush that contains a 4-card straight flush, they’ll pay you 15-to-1 (instead of 9-to-1). With this liberal rules interpretation, the house edge is 3.75%. The total possible outcomes for an optimal player are listed below.

Optimal Play Outcomes (Liberal Rules)

Outcome

Combinations

Frequency

Net

Return

7-card Straight Flush

3,360

2.3919E-07

1002

0.000240

6-card Straight Flush

167,160

1.1900E-05

502

0.005974

7-card Flush

697,620

4.9662E-05

302

0.014998

5-card Straight Flush

4,127,760

0.000294

102

0.029972

6-card Flush

26,945,100

0.001918

22

0.042119

4-card Straight Flush

65,648,544

0.004673

17

0.079447

5-card Flush

372,841,560

0.026542

11

0.291959

4-card Flush

2,627,978,496

0.187080

7

1.309557

Nothing

5,035,629,456

0.358475

-3

-1.075424

Fold before river

4,431,366,576

0.315459

-2

-0.630917

Fold before flop

1,481,973,168

0.105498

-1

-0.105498

Total

14,047,378,800

1.000000

-0.037493

If the rules are interpreted strictly, and you must make a straight flush with all your cards of the same suit, then the house edge is 5.41%.

Optimal Play Outcomes (Strict Rules)

Outcome

Combinations

Frequency

Net

Return

7-card Straight Flush

3,360

2.3919E-07

1002

0.000240

7-card Flush

717,360

5.1067E-05

302

0.015422

6-card Straight Flush

147,420

1.0494E-05

502

0.005268

6-card Flush

27,960,660

0.001990

22

0.043790

5-card Straight Flush

3,112,200

0.000222

102

0.022598

5-card Flush

397,427,940

0.028292

11

0.311212

4-card Straight Flush

41,062,164

0.002923

17

0.049693

4-card Flush

2,627,978,496

0.187080

7

1.309557

Nothing

5,035,629,456

0.358475

-3

-1.075424

Fold before river

4,431,366,576

0.315459

-2

-0.630917

Fold before flop

1,481,973,168

0.105498

-1

-0.105498

Total

14,047,378,800

1.000000

-0.054059

All-Or-Nothing Side Bet

Outcome

Combinations

Frequency

Net

Return

All hole cards same suit

2,860

0.010564

30

0.316927

All hole cards different suits

28,561

0.105498

5

0.527491

Others

239,304

0.883938

-1

-0.883938

Total

270,725

1.000000

-0.039520

Of course, the only reason why I analyzed the game was to Monte Carlo the 6-way collusion edge. The return is about +4.07% for 6 players sharing perfect info under strict rule interpretation, and +5.67% under the liberal rules. That’s not much, considering it’s pretty hard to convey suit information between confederates. It’s probably not worth anyone’s trouble to attack the game. I didn’t bother working out a practical strategy.

(FYI, I’m spending a lot more time outside of the casino these days. Before, I used to practically live in the casino. About 9 months ago, I changed obsessions. You can read about my current mania on my other blog.)

While visiting the TCSJohnHuxley booth at G2E last week, I played at the Lucky Draw Baccarat demo table. It’s a fun game that plays like midi-baccarat, where you can squeeze your draw card. Each player wagers an initial bet, and receives their own 2-card starting hand. Everyone plays against the bank hand, whose first card is exposed. Each player may wager an optional 1x Draw bet to receive a 3rd card, or otherwise stand pat. After the action is complete, the banker reveals his hole card. The bank draws a 3rd card when his two-card total is less than five points. Otherwise the bank stands with 5 points or more.

The game is fun, because winning hands pay odds for drawn 7, 8, and 9 totals. The player makes a decision based on his 2-card total, and the exposed bank upcard. Winning hands pay even-money on the initial wager, and odds on the 1x Draw bet according to the paytable:

Draw Bet Paytable

Outcome

Paytable 1

Paytable 2

Lucky 9 (3-card 9)

3-to-1

3-to-1

Lucky 8 (3-card 8)

2-to-1

2-to-1

Lucky 7 (3-card 7)

2-to-1

3-to-2

6 or less

1-to-1

1-to-1

I analyzed the game to check the house edge, and to run EORs. The outcomes for an 8-deck shoe and optimal player decisions are listed below.

Lucky Draw Baccarat Outcomes (Paytable 1)

Outcome

Combinations

Frequency

Net

Return

Win w/ Lucky 9

134,129,168,192,512

0.053669

4

+0.214675

Win w/ Lucky 8

98,365,258,946,560

0.039359

3

+0.118076

Win w/ Lucky 7

93,724,551,243,776

0.037502

3

+0.112506

Win on draw

260,100,744,978,432

0.104074

2

+0.208147

Lose on draw

976,828,113,772,544

0.390856

-2

-0.781713

Tie on draw

165,885,343,716,480

0.066375

0

+0.000000

Win on stand

471,832,788,590,592

0.188794

1

+0.188794

Lose on stand

195,977,691,906,048

0.078416

-1

-0.078416

Tie on stand

102,355,476,404,736

0.040955

0

+0.000000

Total

2,499,199,137,751,680

1.000000

-0.017931

The house edge for Paytable 1 is 1.79%, and 3.34% for Paytable 2.

The basic strategy for the Paytable 1 game is shown in the table below.

Basic Strategy (Paytable 1)

Total

Upcard

0

1

2

3

4

5

6

7

8

9

9

D

S

S

S

S

S

S

S

S

S

8

S

S

S

S

S

S

S

S

S

S

7

D

D

S

S

S

S

S

S

S

S

6

D

S

S

S

S

S

S

S

S

S

5

D

D

D

D

D

D

D

D

S

S

4

D

D

D

D

D

D

D

D

D

S

3

D

D

D

D

D

D

D

D

D

S

2

D

D

D

D

D

D

D

D

D

D

1

D

D

D

D

D

D

D

D

D

D

0

D

D

D

D

D

D

D

D

D

D

The computed single-card EORs for an 8-deck game with Paytable 1 are fairly low. Still, I checked the countability of an 8-deck shoe, assuming only 15 cards cut off the end. For the simple count below, the game gets advantageous only 3.2% of the time (count is +40 or better), and for an average of only +0.23%/bet. That’s essentially worthless. You might pick up some additional edge with indexed plays, or better yet, using a computer and full knowledge of shoe composition. But overall, this game is surprisingly uncountable, given the options and the odds on the Draw bet.

If I haven’t been posting a lot lately, it’s either because I’m gambling too much, or the edges are too good to post publicly. While both these reasons would normally apply to Galaxy Gaming‘s new Lucky Win Baccarat Side bet, Eliot Jacobson just found out about it, so it’s a free-for-all while it lasts. Hopefully, you can still find a placement in the UK.

I picked up the literature for the game at Galaxy’s booth @ G2E last week. Lucky Win is a baccarat side bet that pays out for wins on low totals. When you bet on Lucky Player, you’re paid when the player wins with 5 points or less. If you bet on Lucky Banker, you’re paid for a banker win with 4 points or less. The top end of the paytable is very nice.

Lucky Win Baccarat Paytable

With With

Lucky Banker(to-1)

Lucky Player(to-1)

1 in Spades

500

500

1 Suited

200

200

1 Offsuit

30

30

2 points

20

20

3 points

12

12

4 points

8

8

5 points

lose

5

The basic house edge is computed in the following tables (8 deck shoe). The Lucky Player has a nominal 12.04% house edge, and the Lucky Banker has a nominal 10.46% house edge.

Lucky Banker Baccarat Side Bet

Outcome

Combinations

Frequency

Payout

Return

Banker win w/ 1 in Spades

373,248,411,648

0.000075

500

+0.037337

Banker win w/ 1 Suited

1,119,745,234,944

0.000224

200

+0.044804

Banker win w/ 1 Offsuit

22,798,126,252,032

0.004561

30

+0.136833

Banker win w/ 2

44,681,581,871,104

0.008939

20

+0.178784

Banker win w/ 3

72,927,778,568,192

0.014590

12

+0.175083

Banker win w/ 4

163,359,790,133,248

0.032682

8

+0.261459

Others

4,693,138,005,032,192

0.938928

-1

-0.938928

Total

4,998,398,275,503,360

1.000000

-0.104629

Lucky Player Baccarat Side Bet

Outcome

Combinations

Frequency

Payout

Return

Player win w/ 1 in Spades

378,622,455,808

0.000076

500

+0.037874

Player win w/ 1 Suited

1,135,867,367,424

0.000227

200

+0.045449

Player win w/ 1 Offsuit

23,124,703,715,328

0.004626

30

+0.138793

Player win w/ 2

44,328,525,111,296

0.008869

20

+0.177371

Player win w/ 3

62,946,423,310,336

0.012593

12

+0.151120

Player win w/ 4

86,165,771,096,064

0.017239

8

+0.137909

Player win w/ 5

122,838,277,197,824

0.024576

5

+0.122878

Others

4,657,480,085,249,280

0.931795

-1

-0.931795

Total

4,998,398,275,503,360

1.000000

-0.120400

The calculated EORs are pretty high, and lend to a very simple unbalanced count. One count nicely fits both the Lucky Player and Lucky Banker bets, for spade and non-spade cards.

Lucky Banker EORs (8 Deck)

Removed

EOR(spade)

EOR(non-spade)

UnbalancedCount

Deuce

-0.065642%

-0.018710%

Trey

0.046125%

0.091348%

Four

0.098900%

0.136220%

+1

Five

0.292038%

0.334057%

+1

Six

0.288705%

0.333087%

+1

Seven

0.254152%

0.296456%

+1

Eight

0.136562%

0.201645%

+1

Nine

0.113699%

0.180474%

+1

Ten/Face

-0.357683%

-0.277855%

-1

Ace

-0.368121%

-0.231719%

-1

Lucky Player EORs (8 Deck)

Removed

EOR(spade)

EOR(non-spade)

UnbalancedCount

Deuce

-0.078258%

-0.033211%

Trey

0.086812%

0.130836%

Four

0.207816%

0.256991%

+1

Five

0.323979%

0.372323%

+1

Six

0.451908%

0.523600%

+1

Seven

0.260662%

0.331159%

+1

Eight

0.201067%

0.260858%

+1

Nine

0.092613%

0.153953%

+1

Ten/Face

-0.462411%

-0.390156%

-1

Ace

-0.340367%

-0.221416%

-1

Using the simple unbalanced count above (Four thru Nine => +1, Ten thru Ace => -1), and starting at 0 for a new shoe (don’t forget to count the burn card!), you should bet both the Lucky Player and Lucky Banker side bets when the count is +34 or better. For an 8-deck shoe with 15 cards behind the cut card, you’ll be able to bet 6.0% of the hands. The Lucky Player bet has an average edge of +14.0%, and the Lucky Banker bet has an average edge of +10.5%. That’s a whopping combined +1.47% edge per dealt hand. That’s insane. You can see how good the bet gets in the graph below.

Normally, I would never post about something this good. But Eliot is posting today, so the cat’s out of the bag. My apologies to any APs already hitting this game 😦

For now, Viejas is offering a liberal interpretation of the Free Bet Blackjack rules, where they’ll now let you free double on two card soft 19, 20, and 21. The rules printed on the felt say free doubles only on hard 9, 10, and 11, but they’ve interpreted A-8 to mean hard 9, A-9 to mean hard 10, and A-Ten/Face to mean hard 11. This is only slightly helpful to the player.

The Viejas rules are:

Free double on two card hard 9, 10, 11

Free double on two card soft-19, soft-20, and Blackjack

No re-split of Aces

No surrender

Free double after free split

The house edge is 0.88%, which isn’t too bad. Without the free doubles on the soft totals, the house edge would be 1.10%. I guess the free doubles on the soft totals make up for the lack of surrender. The basic strategy for the non-free-split hand is listed below. Refer to the original post for the basic strategy for free-split hands.